atTop_div_const — Mathlib

English

In any nontrivial poset, the top and bottom filters are disjoint: atTop and atBot have empty intersection as filters.

Русский

В любом не тривиальном частично упорядоченном множестве верхний и нижний фильтры несовместимы: atTop и atBot не имеют общих элементов.

LaTeX

$$$\mathrm{Disjoint}(\mathrm{atTop}, \mathrm{atBot})$$$

Lean4

/-- If a function `f` tends to infinity along a filter, then `f` divided by a positive
constant also tends to infinity. -/
theorem atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) : Tendsto (fun x => f x / r) l atTop := by
  simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)

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